Properties

Label 1089.6.a.p
Level $1089$
Weight $6$
Character orbit 1089.a
Self dual yes
Analytic conductor $174.658$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1089.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(174.657979776\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 7) q^{2} + ( - 13 \beta + 25) q^{4} + ( - 10 \beta - 24) q^{5} + ( - 62 \beta - 42) q^{7} + ( - 71 \beta + 55) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 7) q^{2} + ( - 13 \beta + 25) q^{4} + ( - 10 \beta - 24) q^{5} + ( - 62 \beta - 42) q^{7} + ( - 71 \beta + 55) q^{8} + ( - 36 \beta - 88) q^{10} + (74 \beta + 28) q^{13} + ( - 330 \beta + 202) q^{14} + ( - 65 \beta + 153) q^{16} + (372 \beta - 550) q^{17} + (852 \beta - 12) q^{19} + (192 \beta + 440) q^{20} + (330 \beta - 46) q^{23} + (580 \beta - 1749) q^{25} + (416 \beta - 396) q^{26} + ( - 198 \beta + 5398) q^{28} + ( - 1492 \beta + 1094) q^{29} + (1600 \beta - 6040) q^{31} + (1729 \beta - 169) q^{32} + (2782 \beta - 6826) q^{34} + (2528 \beta + 5968) q^{35} + ( - 2816 \beta + 454) q^{37} + (5124 \beta - 6900) q^{38} + (1864 \beta + 4360) q^{40} + ( - 8 \beta + 18246) q^{41} + (3112 \beta - 6440) q^{43} + (2026 \beta - 2962) q^{46} + (390 \beta - 22066) q^{47} + (9052 \beta + 15709) q^{49} + (5229 \beta - 16883) q^{50} + (524 \beta - 6996) q^{52} + (7102 \beta + 2536) q^{53} + (3974 \beta + 32906) q^{56} + ( - 10046 \beta + 19594) q^{58} + ( - 1980 \beta + 2384) q^{59} + ( - 2026 \beta + 13664) q^{61} + (15640 \beta - 55080) q^{62} + (12623 \beta - 19911) q^{64} + ( - 2796 \beta - 6592) q^{65} + ( - 12704 \beta - 13908) q^{67} + (11614 \beta - 52438) q^{68} + (9200 \beta + 21552) q^{70} + (4354 \beta - 17870) q^{71} + ( - 5568 \beta + 26174) q^{73} + ( - 17350 \beta + 25706) q^{74} + (10380 \beta - 88908) q^{76} + ( - 11426 \beta + 14138) q^{79} + (680 \beta + 1528) q^{80} + ( - 18294 \beta + 127786) q^{82} + (21960 \beta + 28740) q^{83} + ( - 7148 \beta - 16560) q^{85} + (25112 \beta - 69976) q^{86} + ( - 26704 \beta + 40454) q^{89} + ( - 9432 \beta - 37880) q^{91} + (4558 \beta - 35470) q^{92} + (24406 \beta - 157582) q^{94} + ( - 28848 \beta - 67872) q^{95} + (9924 \beta - 125746) q^{97} + (38603 \beta + 37547) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 13 q^{2} + 37 q^{4} - 58 q^{5} - 146 q^{7} + 39 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 13 q^{2} + 37 q^{4} - 58 q^{5} - 146 q^{7} + 39 q^{8} - 212 q^{10} + 130 q^{13} + 74 q^{14} + 241 q^{16} - 728 q^{17} + 828 q^{19} + 1072 q^{20} + 238 q^{23} - 2918 q^{25} - 376 q^{26} + 10598 q^{28} + 696 q^{29} - 10480 q^{31} + 1391 q^{32} - 10870 q^{34} + 14464 q^{35} - 1908 q^{37} - 8676 q^{38} + 10584 q^{40} + 36484 q^{41} - 9768 q^{43} - 3898 q^{46} - 43742 q^{47} + 40470 q^{49} - 28537 q^{50} - 13468 q^{52} + 12174 q^{53} + 69786 q^{56} + 29142 q^{58} + 2788 q^{59} + 25302 q^{61} - 94520 q^{62} - 27199 q^{64} - 15980 q^{65} - 40520 q^{67} - 93262 q^{68} + 52304 q^{70} - 31386 q^{71} + 46780 q^{73} + 34062 q^{74} - 167436 q^{76} + 16850 q^{79} + 3736 q^{80} + 237278 q^{82} + 79440 q^{83} - 40268 q^{85} - 114840 q^{86} + 54204 q^{89} - 85192 q^{91} - 66382 q^{92} - 290758 q^{94} - 164592 q^{95} - 241568 q^{97} + 113697 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.37228
−2.37228
3.62772 0 −18.8397 −57.7228 0 −251.081 −184.432 0 −209.402
1.2 9.37228 0 55.8397 −0.277187 0 105.081 223.432 0 −2.59787
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.6.a.p 2
3.b odd 2 1 363.6.a.f 2
11.b odd 2 1 99.6.a.d 2
33.d even 2 1 33.6.a.e 2
132.d odd 2 1 528.6.a.o 2
165.d even 2 1 825.6.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.e 2 33.d even 2 1
99.6.a.d 2 11.b odd 2 1
363.6.a.f 2 3.b odd 2 1
528.6.a.o 2 132.d odd 2 1
825.6.a.c 2 165.d even 2 1
1089.6.a.p 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1089))\):

\( T_{2}^{2} - 13T_{2} + 34 \) Copy content Toggle raw display
\( T_{5}^{2} + 58T_{5} + 16 \) Copy content Toggle raw display
\( T_{7}^{2} + 146T_{7} - 26384 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 13T + 34 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 58T + 16 \) Copy content Toggle raw display
$7$ \( T^{2} + 146T - 26384 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 130T - 40952 \) Copy content Toggle raw display
$17$ \( T^{2} + 728 T - 1009172 \) Copy content Toggle raw display
$19$ \( T^{2} - 828 T - 5817312 \) Copy content Toggle raw display
$23$ \( T^{2} - 238T - 884264 \) Copy content Toggle raw display
$29$ \( T^{2} - 696 T - 18243924 \) Copy content Toggle raw display
$31$ \( T^{2} + 10480 T + 6337600 \) Copy content Toggle raw display
$37$ \( T^{2} + 1908 T - 64511196 \) Copy content Toggle raw display
$41$ \( T^{2} - 36484 T + 332770036 \) Copy content Toggle raw display
$43$ \( T^{2} + 9768 T - 56044032 \) Copy content Toggle raw display
$47$ \( T^{2} + 43742 T + 477085816 \) Copy content Toggle raw display
$53$ \( T^{2} - 12174 T - 379065264 \) Copy content Toggle raw display
$59$ \( T^{2} - 2788 T - 30400064 \) Copy content Toggle raw display
$61$ \( T^{2} - 25302 T + 126184224 \) Copy content Toggle raw display
$67$ \( T^{2} + 40520 T - 921013232 \) Copy content Toggle raw display
$71$ \( T^{2} + 31386 T + 89872392 \) Copy content Toggle raw display
$73$ \( T^{2} - 46780 T + 291320452 \) Copy content Toggle raw display
$79$ \( T^{2} - 16850 T - 1006085552 \) Copy content Toggle raw display
$83$ \( T^{2} - 79440 T - 2400814800 \) Copy content Toggle raw display
$89$ \( T^{2} - 54204 T - 5148586428 \) Copy content Toggle raw display
$97$ \( T^{2} + 241568 T + 13776267004 \) Copy content Toggle raw display
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